wkb treatment of laser radiation in the transient unstable region

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Physica 107A (1981) 71-90 North-Holland Publishing Co. WKB TREATMENT OF LASER RADIATION IN THE TRANSIENT UNSTABLE REGION Toshihico ARIMITSU* Department o[ Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113, Japan Received 10 June 1980 Revised 6 October 1980 The transient behavior of laser radiation from the unstable state (i.e., no photon at the initial time) is treated by the WKB treatment proposed by Caroli et al., and comparison of the results with those of the scaling theory is given. A simple approximation, needed for the WKB treatment to obtain the distribution function, is also given. 1. Formulation Recently, Caroli, Caroli and Roulet 1"2) showed that, in the WKB region, Suzuki's scaling distribution function was obtained by the WKB treatment of the "Schr6dinger equation", equivalent to the Fokker-Planck equation for the problem of diffusion in a bistable potential in the limit of a small diffusion constant when the system starts from the unstable point. In this treatment "configuration space" is divided into several regions, and the wave functions for different regions are matched by the WKB method. This aspect of the transient unstable problems is attractive especially because it may be possible by this method to extend the scaling theory 3~) to the field theory7). In this paper we apply the WKB method to the laser systemS), when it starts from the unstable point, which corresponds physically to the situation that the laser system is suddenly switched on above threshold, and consequently that there is no photon initially in the cavity, in order to compare the results obtained by the scaling theory 9'1°) and to investigate the relation between the scaling theory and WKB formalism. We start the analysis with the dimensionless Fokker-Planck equation in the polar coordinates, aft" l . 1 +r~r{( l _r2)r2~ , ?¢}=Or l O Or[ Or J ~-~-Y' (1.1) .-£ Ot which corresponds to eq. (2.9) of ref. 9. The smallness parameter, 0, is given by E K (N:)~ 1 0 = ~ = 2 -- (1.2) '~l[ ~ (or0 -- O'th)2' *Present address: Institute of Physics, University of Tsukuba, Ibaraki, 305, Japan. 0378-4371/81/0000-0000/$2.50 O North-Holland Publishing Company 71

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Physica 107A (1981) 71-90 North-Holland Publishing Co.

WKB TREATMENT OF LASER RADIATION IN THE TRANSIENT UNSTABLE REGION

Toshihico ARIMITSU*

Department o[ Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113, Japan

Received 10 June 1980 Revised 6 October 1980

The transient behavior of laser radiation from the unstable state (i.e., no photon at the initial time) is treated by the WKB treatment proposed by Caroli et al., and comparison of the results with those of the scaling theory is given. A simple approximation, needed for the WKB treatment to obtain the distribution function, is also given.

1. Formulation

Recently, Caroli, Caroli and Roulet 1"2) showed that, in the WKB region, Suzuki's scaling distribution function was obtained by the WKB treatment of the "Schr6dinger equation", equivalent to the Fokker-Planck equation for the problem of diffusion in a bistable potential in the limit of a small diffusion constant when the system starts from the unstable point. In this t reatment "configuration space" is divided into several regions, and the wave functions for different regions are matched by the WKB method. This aspect of the transient unstable problems is attractive especially because it may be possible by this method to extend the scaling theory 3~) to the field theory7).

In this paper we apply the WKB method to the laser systemS), when it starts from the unstable point, which corresponds physically to the situation that the laser system is suddenly switched on above threshold, and consequently that there is no photon initially in the cavity, in order to compare the results obtained by the scaling theory 9'1°) and to investigate the relation between the scaling theory and WKB formalism.

We start the analysis with the dimensionless Fokker-Planck equation in the polar coordinates,

aft" l . 1 +r~r{(l_r2)r2~,?¢}=Orl O Or[ Or J ~ - ~ - Y ' (1.1) .-£

Ot

which corresponds to eq. (2.9) of ref. 9. The smallness parameter, 0, is given

by E K ( N : ) ~ 1

0 = ~ = 2 -- (1.2) '~l[ ~ (or0 -- O'th)2'

*Present address: Institute of Physics, University of Tsukuba, Ibaraki, 305, Japan.

0378-4371/81/0000-0000/$2.50 O North-Holland Publishing Company

71

72 T . A R I M I T S U

where ~r0 is the pumping parameter, O' th = r3,±/gZN, 3'11 and 3'1 represent the longitudinal and transverse relaxation rates, respectively. In deriving (1.1), one used the model of the running-wave, single-mode ring laser with N identical two-level homogeneously broadened atomsS). Then the pumping parameter is written in terms of the effective temperature, TA, of the reservoir for atom systems as

E t r o = - t h 2 k B T A, ( - l ~ t ~ 0 ~ l ) , (1.3)

where e represents the energy difference of the relevant two levels. Further- more, (Nz)t~ in (1.2) denotes the number of atoms in the upper level at t ~ . It should be noticed that the definitions of ~r0 and (rth are different from those in ref. 9. Eq. (1.1) is valid for the laser operating near thresholdH). On the other hand, we are interested in the small 0 cases, which correspond to strong pumping. Then we can estimate the range of o-0 for the following treatment as

~ ] ~ ~ ~r0 - cth ~ 1. (1.4)

It should be noted that, in an ordinary laser, the condition K ~ 711 is satisfied, 1:'~3) and that the right-hand side of (1.4) is proportional to the inverse of the square root of the number of atoms. We may obtain the pumping intensity satisfying the condition 0 .4) for the pumping not too strong.

The stationary solution of (1.1) is given by

lZV's(r) = N___ e-V~/°, (1.5) 2~r

where

I _ 1 r2 ' U(r) = ~ r 4 (1.6)

N-~=fdrre -U(" / °=X/Oe ' /a ° {X / - -~+er f (2~) }2 , (1.7) 0

with x

= [ ds e -~:. eft(x) 0

In the following in this section we omit the hats over W and t for simplicity. We can obtain the solution of (1.1) in the following form ~4'~5)

W(r, q~, t) = Y~ Y~ A.., qjrr----~) tk..,(r) e '"~ e A.,.,o, (1.8) m = 0 n = - ~

WKB TREATMENT OF LASER RADIATION 73

with

~boo(r) = ~ e-°l~°}u{% )too = 0, (1.9)

w h e r e q&m(r) a n d A.~. are the e i g e n f u n c t i o n s a n d the e i g e n v a l u e s , r e s p e c t i v e l y ,

of the " S c h r f d i n g e r " e q u a t i o n :

- - 0 2 dEll&m _ _

~ - r [ V . ( r ) - A.,.]~b.,. = 0 , (1.10)

with

2 tl

V,(r) = ,, [ r 2 - ~boo(r)J

1 1 I ( 1 _ 8 0 ) r 2 _ ~ r +.~ : 02(/12 -- ~ ) r--~ + 0 d- 1 4 l r6" ( l . l l )

The schematic shape of V,(r) is given in fig. 1. Using the orthogonality of the eigenfunctions of the "Schr6dinger" equation (1.10),

f dr @.m(r)O.m,(r) = 8m~,, 0

(1.12)

V(r)

/ V1(r) ,/ \ VIH(r) I /

o.o~ \ , \ / / /

/ J

0.02

0

- 0.02

(i)

T 0.5 T 1.o

> • (n l ) ) --- ( I I ) • • ( IV)

}P

r

Fig. l. Schematic shape of V,=o(r) defined by (1.11); solid line. 7} and ~ are the boundaries where, for r<-q, the potential V.=o(r) can be well approximated by V[(r), (1.18), and for ~ < r the potential can be approximated by Vm(r), (1.19). V,=o(a) = V,=o(/3) = V,=o(/3') = A.

74 T. ARIMITSU

and their completeness

8(r - r') = f~ ~, , .(r)tk, m(r'), (1.13) r n=0

1 8(~ - q:') = ~ , _ ~ e i"(~ ~'), (1.14)

the Green function of the Fokker -P lanck equation (1.1), with the initial condition:

1 8(r - r')8(~ -~'), (1 .15 ) W ( r , ~, t = O) = r

is given by 14A5)

WG(r, q~" r', q~'; t) 1 Ooo(r.__...~) + ~ O,~(r)q&m(r') e i"w *')e a°mt/o ' 2wr qJoo(r') ~-o . . . . "

The solution of the Fokker -P lanck equation (1.1) is given by

2w

W ( r , , c , t ) = f dr'r'f d~' WG(r, 0 0

,;; r', ,;'; t ) W ( r ' , ,p', t = 0).

For the unstable initial condition

W ( r , ~, t = 0) = lim ~ r 6(r - A),

(1.16)

(1.17)

1 W ( r , ~, t) = ~ Ooo(r) £ A, .~om(r) e -~°m'/°,

r t l=0

where

Am = lim $°'(A) (1.20)

As we confine ourselves to this unstable situation in this paper, the dis- tribution function does not depend on the phase q~ as seen from (1.19), so that it is sufficient to solve the "Schr6dinger" equation for n = 0, i.e., s-wave eigenfunctions and eigenvalues. It is consistent with the assumption used in ref. 9). We will omit the subscript n hereafter in this paper.

(1.19)

in which we are interested, the transient distribution function is given by ~5)

(1.18)

WKB TREATMENT OF LASER RADIATION 75

2. W K B a p p r o x i m a t i o n

Following Caroli, Caroli and Roulet~'2), we solve the "Schr6dinger" equa- tion (1.10) in terms of the WKB approximation by dividing the "configuration space" into four regions as seen in fig. 1, in order to obtain the eigenfunctions of the "exci ted" states. The ground state eigenfunction is already given by (1.9). We approximate the potential (1.11) as

02 1 1 Vi(r) = - - ~ - ~ + 0 + (1 - 8 0 ) r 2, (2.1)

Vm(r) = -O ( l + l---~ 0 ) + ( 1 + 160 )(r - ro) 2, (2.2)

in the regions (I) and (III), respectively, where r0 = 1 + 20 + (~(02). In region (I), we can solve the "Schr6dinger" equation (l.10) by ap-

proximating the potential V,(r) by Vi(r) as

~bi(r) = KI X,/r exp { - ~'k,/1 ----S"~ } L ~) (~X,/1 --'-S"~), (2.3)

where L~)(z) are the Laguerre functions which satisfy the differential equa- tion

d2U dU z --d~-z2 + (1 - z) --d-~-z + vU = 0 (2.4)

To obtain (2.3), we have used the boundary condition that ~ ( 0 ) = 0, because V,(r) has the singularity at r = 0. The "eigenvalue" of this regime is related to v by the relation

Av = 0{1 + 2~/1 - 80(v + ½)} = 2(v + 1)0 + (?(02). (2.5)

In region (II), the solution of (1.10) is approximated by the WKB method as r

0 x] IK"exp[Jo VV(r')-x ~lI(r) = N/ V( r )_

r

dr'

where c~ is the turning point between the regions (I) and (II) defined by V(a) = A (see fig. 1). The coefficients Kn and Kh are related to KI by the matching procedure of the WKB method, which is given in appendix A, as

. 1 /~ / ] -~ -~ '~ \ - (v+t ) r e ~ / 1 80 1-(v+1/2) = KILN/2[ --------~- 1 | e l]4g(v)-'n'/S' --/~'I~----~V)~" 2- ") L~/(2v + 1) + ~3 (2.7)

76 T. ARIMITSU

cos v+ /vl- 8o) + i,v [ e V l - 80 ]++,,2 I ~ ~ " 2- = K LX/(-~ + ~ + ~3 e-""g'~'++'8' (2.8)

with

sin_l[ 2v + 1 ] g(v) = [X/(2-~-+ 1~2 + ~j. (2.9)

In region (III), using (2.2) we obtain the solution of (1.10) as

, 2 - t o ) ) , + i , , ( r ) = K m D ~ ( - q 2 V l + 160(r-ro))+ KillD,u.(q-~/l -I- 160(r /

(2.10)

where D.(z) are the Weber functions which satisfy the differential equation

--d~- + / x + ~ - U = 0 . (2.11)

The coefficients Knl and K}u are related to KH and Kh by the matching procedure given in appendix A as

[ ")4Q \ 1 1 4 1 ~ \ I / 2 ( p . + 112)

Klll+KillCOSl~,rr=KlleS=,(hll ~ | [ - - - ~ ) , (2.12) \X/1 + 1607 \t~ + ~

" ' ~ K . " 20 . , 4 . _1_1\1/2(p++1/2) = ' e-So. <*>/' _ _ ~ l t+ ~-++) , (2.13)

~ m F ( - p . ) \ ~ / 1 + 160] \ e ]

with

dr S~(X) = f --~- v'v/--V-~ - ,~, (2.14)

cl

where /3 is the turning point between the regions (II) and ( l ID defined by V(/3) = h (see fig. I). The relation between the "eigenvalue" of this regime and /.t is given by

A ~ , = 0 { - - ( 1 + 7 0 ) + 2 ~ ( ~ + + ) } = 2 p 0 + (/(0) 2" (2.15)

In the region (IV), the solution of (1.10) is given in the WKB approximation as

r

° ]'2exp[ fT V r) 2,6> qqv(r) = K[v[x/V(r ) _ X

W

Here /3' is the turning point between the region (III) and (IV) defined by V(/3') = h (see fig. 1). The matching procedure of the WKB method gives the

WKB T R E A T M E N T OF LASER RADIATION 77

relations between K~v, Kul and K~n as

Ki l l = 0 ,

K' [ 20 ~1]4[ e ~ ( l~+~) - IV ' ~ ~ -- K ~il + ~Vl + 160] ~+~}

K n l COS /~'rr,

(2.17)

(2.18)

where (2.17) relates to the fact that, in the region (IV), we have omitted the growing part for large r in (2.16).

From (2.7), (2.8), (2.12), (2.13), (2.17) and (2.18), all the coefficients, Kll, Kil, Kin, Kill and Kiv, are related to KI. Then we obtain the condition, which determines the eigenvalues, as

I F ( - v ) F ( - I x ) l x / ( 2 v + 1)2 + ~ J \ Ix + ½ ]

[ 2e ]-(~+½) e ~g(~)-~/4 = 0. (2.19) 1 + e-2S~'~) cos w- cos t~r F(1 + v---~ LV(2v + 1)2 +~J

The parameters v and p~ are related to A by the relations (2.5) and (2.15), respectively. With these relations, the condition (2.19) determines the eigen values, A, of the "Schr6dinger" equation (1.10), with the potential (1.11), for n = 0 and small 0 limit (WKB limit). As the gaps between eigenvalues are of the order of 0 whereas the extremum of the potential is V(rmax) = (1/27)(1 + 90) with rmax -- (1/3)(1-- 60) for small 0, S~,(A) defined by (2.14) is large for the eigenvalues, A ~ V(rma0, which play dominant roles in the unstable case, so that we can neglect the second term of (2.19) compared to the first term. Then the solutions, it, of (2.19) are approximated by it~ and it~ given by (2.5) and (2.15), respectively, for v being zero and positive integer and tx being positive integer. In the following, we assume that none of the eigenstates, which are the solutions of (2.19), is degenerate. Then the "exci ted" states, which are well approximated by it~ with v =0 , 1,2 . . . . . and it~ with Ix = v + 1 = 1, 2, 3 . . . . . appear in sequence.

The wave functions which correspond to these "exci ted" states are obtained as follows. For the states v =p (p = 0, 1, 2 , . . . ) , the wave functions are localized in the region (I) and extend to the regions (II), (III) and (IV) by the "tunneling" effect. Using the relations between the coefficients of each region, (2.7), (2.8), (2.12), (2.13), (2.17) and (2.18), we obtain the wave functions for each region as

~ ~/1 ---:--~ } L ~°) (~-~X/]-L-'~), (2.20a) 0}l~(r) = Kl~ / r exp{_ 2 r 2

78 T. ARIMITSU

• 1 . l /2r 0 ]1/2

r

x{exp[-f dr' " Xo] -~-V V(r') -

r [/~ + ~\"+~ I- f dr' .

+ e zso0al x cos t x T r / - - I e x p l / - - V V ( r ' ) - \ e / k J 0

I (1, _ K / | ~1]2{ 2 0 ) ' / ' r ( - t~)(be + ½) ~''+'~'

×e s"~"'G(p)D~(~/2X/l + 160(r- ro)),

It) / 1 \ I / 2 F ( - - g ) e s"o~z~

r

)t o J t3'

where

(2.20b)

(2.20c)

~o ], (2.20d)

e4~,cp~_~/8 ( - )P f~/(2p + I)2+~] °+~ G(p) - - F ( 1 + p ) [ 2 e ' ( 2 . 2 1 )

X o = 0{1 + 2%/1-80(p +~)}= 20(p + 1), (2.22)

and ~ is related to p by the relation

tx =(p + 1 ) - ( 1 2 p + ~ ) 0 + 6(02), (2.23)

which is obtained by putting X~, (2.15), equal to Xp, (2.22). Because the wave functions (2.20) are well localized in the region (I), we can obtain K~ by the normalization condition, (1.12), in which we approximate the wave functions by (2.20a) for all the regions;

KI = (2.24)

For the states ~ = q (q = 1, 2, 3 . . . . ), the wave functions are localized in the region (liD. By using (2.7), (2.8), (2.12), (2.13), (2.17) and (2.18), the wave functions for each region are given as

WKB TREATMENT OF LASER RADIATION 79

! l

.i.(lll)[.~_ X/1 160 ,/4 2(q+2) ~,,,q ~-j - Ki, e s°~(~) e~g~v)-~/s( - )q+'

F(_v)[ .V ' (2 v 2 ,_ ,+½ 2 2 X • ~e 1)+4] ~ / r e x p { - ~ lX/]-Z~-80}L~'(~-~oVTZ~),

(2.25a)

II'q''' = ~IIIL 20 ] e t- - 7 / LVV(r)- Ao J r

× { exp [I ] Aq F(1F(- v~) c°s + wr [.V(2v + 1)2 +~] 2(v÷½~ 2e a

r

xe'~'~'-~"exp[f yVV(r ' ) - .~ , (2.25b) a

re,q,-) - +

,Ira (~/I_+ 160~"4/q + ½~ '(q+½' ~b,v.q(r)=K[,, 20 ] \ e ]

r

× [vv(;)- ~J

where

(2.25d)

0{ ,226,

and v is related to q by the relation

v = ( q - 1 ) + ( 1 2 q - 1 ) O +~(02). (2.27)

As the wave functions (2.25) are well localized in the region (III), Kh, is given by the normalization condition, in which the wave functions are approximated by (2.25c) for all the regions, as

K hi X/~--~.v\ w 0 / . ( 2 . 2 8 )

80 T. ARIMITSU

3. Distribution function and moments

3.1. Comparison with the scaling theory

In the WKB approximation, the transient distribution function for the unstable initial condition, (1.19), reduces to

W(r, q~, t) = Ws(r) + w(r, q~, t), (3.1)

with W~(r) given by (1.5) and

1 w(r, q~, t) a--7_qJoo(r)(Y, (l) m e-~,/o + <, Ap ~bp (r) z~ (iil) (11i) . ~ -- Aq ~bq ( , ~ ~ ~, (3.2) /.,n-r I.p=0 q=l

where ~b~)(r) and ~m)(r) are given by (2.20) and (2.25), respectively, cor- responding to the regions to which the argument r belongs, and Ap and Jtq are approximated by (2.22) and (2.26), respectively; q, oo(r) is given by (1.9). By substituting (2.20a) and (2.25a) into (1.20), A~ I) and a (nn ,.q are obtained explicitly by

(I) A~ l)= lira 0P~)(A) = lim 0"P(A) - K~

~o x / ~ ,,~0 V ~ - X V N - ' (3.3)

d, tm)(A~ a.(nl)tA~ A~ ul)= lim vq , - . = lim ~VLq t~/

aoo X/NA a-o V/N--A

_ K~i,(3/l+160)"4e-~o,~, :

( v ) - [ + '<_ q + _~ ~(o*~) X/(2v + 1) 2 4[ X (__)q+ 1 F ( - - V ) 2e (3.4)

respectively, where v is given by (2.27), K~ and Kh~ are given by (2.24) and (2.28), respectively. We have used the fact that A is in the region (I) and that L~°)(0) = 1. It should be remarked that w(r, 4), t) has the following important characteristics:

2rr

f drr f (3.5) o o

which can be easily seen from (1.19) by using the orthogonality condition (1.12).

For r in the region (I), substituting (2.20a) and (2.25a) into (3.2), we obtain

K : wE(r, q~, t) - .x~ e (~/2o)v(,) (r2/4O)~, l-/TT~gO -2rr

× ~, L~°'[ r2 Vl_--7~)e-.2,,.i,, ,=o \2-0 + U ( ~ , e-2s~'~')

W K B T R E A T M E N T O F L A S E R R A D I A T I O N 81

= ~ e_~,/s0 1 e_,~/2t~_0~, (3.6) 2~r r - 0

where we have neglected the terms of the order of exp[-2S,~()tq)], and have defined

T = 0 e ~. (3.7)

We have used the generating function for the Laguerre polynomials given by

e -zs/(1-s) = Isl < 1. (3.8)

n=0

If r in the region (I) is so small that we can safely put r4/80 ~-0 even in the limit 0 ~ 0, then (3.6) reduces to

1 e-r2/2('-°), (3.6') w,(r, q~, t) = 2Ir(~" - 0)

which is formally identical to the distribution function of the initial time regime of the scaling theoryg). It should be remarked that the distribution function R(x, t) in ref. 9 is related to W(r, 4~, t) by the relation.

21r

l I R(xo t) = ~ dq~ w(r, ~, t), (3.9) o

with x = r 2.

For r in the region (II), substituting (2.20b) and (2.25b) into (3.2), we obtain

__1 (1 - 2)3,2 f f ( - ) p [ ( 1 - n2)r2 ] p WII(P" ~ t) e- ,# /80 + ~y(e-S~t~)

q~' 2~r r(1 - rE) 2 ~='=o P! t2~r(-i -~r-~J

z * 2x3/2 1 e_~4/80~1 - - ~[~ J e_(l_.o2)r2/2,r(l_r2),

= 2---~ r(1 - r2) 2 (3.10)

where we have neglected the terms of the order of exp[-S~()k)] , and evaluated ~t~!p(r) approximately in the region (II), the details of which are given in appendix B with the definition of rl. As "0 ~ 0x(~<x <½), which follows from the fact that, for r <-O, the potential V(r) can be well ap- proximated by V i ( r ) , ( 3 . 1 0 ) reduces in the limit 0 ~ 0 to

w n ( r , q~, t ) = 1 e_r2/2~O_,2) 27r'r(1 - r2) 2 ' (3.10')

which is formally identical to the distribution function in the scaling regime of the scaling theory 9) for e ~ 0 and T, ~ 0 with r fixed.

82 T. A R I M I T S U

The essential difference between (3.6'), (3.10') and the scaling distribution functions in the initial and scaling regimes 9) is that (3.6') and (3.10') are correct for all t />0 within the approximations mentioned above in the restricted regions of the "configuration space", i.e., in the regions (I) and (II), respec- tively, whereas the scaling distribution functions 9) are correct for all r/> 0 in the restricted regimes of time, i.e., in the initial and scaling regimes. We have neglected Ws(r) in (3.1) in the comparison of (3.6'), (3.10') with the scaling results9), because the contribution of Ws(r) to W(r) in the regions (I) and (II) is very small in the limit 0 ~ 0.

3.2. An approximation to the distribution function in the remaining regions

Now we have to obtain the distribution function in the remaining regions (Ill) and (IV) in order to complete the distribution function. Using (3.6) and (3.10), we can write (3.5) as

2rr n

f ~ [ f drrwl(r,q~, t)+ f drrw.(r,,p, t)+ f drrwu(r,q~, t)] =0 , (3.11) 0 o ~ t~

w h e r e WR(r, tip, t) represents w(r, ti, t) in the remaining regions (III) and (IV); and ~ are the boundaries where, for r <-0, the potential V(r) can be well

approximated by V~(r), (2.1), and, for ~ < r (<~') the potential V(r) can be approximated by Vm(r), (2.2), respectively (see fig. 1). In our unstable initial condition, (1.18), WR(r, t) should satisfy the conditions:

and

WR(r, q~, t = O) = - - W s ( r ) , (3.12)

WR(r, ~, t = ~) = 0. (3.13)

For the first approximation, we a s s u m e WR(r, ti, t) in the form

WR(r, q~, t) = - B(r)Ws(r), (3.14)

where r is related to t by the relation (3.7). Then the conditions (3.12) and (3.13) can be written in terms of B(T) as

B(r = 0) = 1, (3.15)

and

B(r = ~) = O. (3.16)

Substitution of (3.6), (3.10) and (3.14) into (3.11) gives the expression of B(-r)

WKB TREATMENT OF LASER RADIATION 83

as

B (r) = NoD" - 0 e °12('-°~2[erf(so) - erf(s 0]

+ (1 - r12) t/2 e-"/s°-n2/:'[1 - e-(~2-n2)/2~O-~z)]},

where

So = s~ + n 2 / 2 V ~ , st = X/0/X/2(~" - 0),

N o' = e-'/4°N-' = ~/-O { X/~ + erf(2-~ ) },

N I l = V ~ + e r f ( 1 - ~ ' ) / . \ 2 V O / 1

Then moments are given by oo 2rr

l(2)=(r2)= f drr f d~or:W(r,~o,t) 0 0

= I ~:) + I t 2) ± "(2) ± ~ (2) T i l l T / R ,

oo 2~"

I(4)=(r4)= f drr f dq~r4W(r,q~,t) 0 0

: I ~4) d- I ~4) TIII~- i" (4) T~- I ~),

where we have defined 2~r

0 0

7 I 2~r

I} ") = f drr f dq~ rnwi(r , q>, t ) ,

0 0

2~r

It~ )= f drr f dq~r"wll(r,q~,t), 0

21r

i~n)= f drr f dq~ rnwR(r, % t) o

The moments (3.20) and (3.21) are explicitly given by

/~2) = I + ONo e -~/4°,

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22a)

(3.22b)

(3.22c)

(3.22d)

(3.23a)

84 T. A R I M I T S U

and

•{2)= 20 [ 1 - e ,#/8o ~212(r 0)] T - - O

( 2 0 ) 3/2 (~ _-- ~2 Ct[erf(so) - eft(sO],

(2) e-~4/80 nz/2T(1 _ { l I| = T/2) 1/2 - e (c2 -l)/2"r

- - ( 1 - - r 1 2 ) [ E ( 2 " ~ 7 ) - ~ 2e-<';2 i'/2"E(~'2'~]~'\2-~T]JJ

l ~ ) = -NoB('c){N T' + 0 e '~ ' ~)~/4o},

1~4) =

114~ =

i(4) I1 =

1 + 20 + 0 e ~/4o,

4(20)3/2C2'{(s~+1)

1 e ~}, - 21-(So - 2s3 e ~ - ~sl

e -n4/4° n2/2r(1 -- T/2)1/2{1 -- e (~2 I)/2T

-2(1-~2)[E(1) - '2e ';~ W2"E("S'2~]\2"r/J

,>,2,(, "(7))}}' I~ ~ = -NoB(~'){NTI(1 + 20) + 0(1 + ~2) e ~' ~2):/4o},

W

(3.23b)

(3.23c)

(3.23d)

(3.24a)

(3.24b)

(3.24c)

(3.24d)

10

t= 1.5

3.5

0.5 Fig. 2(a)

(a)

1.o r 2

WKB TREATMENT OF LASER RADIATION 85

W

0

W

0

(b )

t = 4 0

(c)

4 - 5

0.5 1.0 r 2

(d) t -ss

(e) 6,5

7 . 5

, , ).

0.5 1-0 r 2

Fig. 2. Time evolution of the distribution function derived by the WKB approximation for typical values of t: a) t =1.5, 2.5, 3.5, b) t =4.0, c) t =4.5, d) t =5.5, e) t =6.5, f) t =7.5. (a =50, rl = 0.065 and ~ = 0.9422272).

where ¢2 = (1 - ~/2)/(1 - ~ 2 ) and E(x) is defined by

= x C I d s e-~s-~. E(x) (3.25) x

In fig. 2, the time evolution of the distribution function, (3.1) with (3.6), (3.10) and (3.14), is given for typical values of t. The parameters rl and ~, which are the boundaries of the regions (see fig. 1), are determined so as to satisfy the condition (3.12), and the condition that the distribution function

86 T. ARIMITSU

should not be negative in any region. We have put "O =0.065 and ~ = 0.9422272 for a = 50. In fig. 3, the first and second cumulants of r 2, (3.23) and (3.24), are given for the same parameters as in fig. 2. The discussions about

them are given in the next section.

4. Discussion

In this paper, we have treated the laser model 8) by the WKB ap- proximation"2), and compared the results with those obtained by the scaling theory9). From the study of the WKB approximation, we know that the distribution function obtained by the scaling theory 9) corresponds to the approximate evaluation of w(r, ch, t) defined by (3.1) and (3.2). Then in the regions where Ws(r) is negligible for all times (I and II for the laser model), we can obtain the same formal expressions of the distribution function as those of the scaling theory. In the regions III and IV of the WKB treatment and in the final regime of the scaling treatment9), we need another ap- proximation to obtain the distribution function of the corresponding regions or regime. For the scaling theoryg't°), we used the l)-expansion theory to obtain the distribution function in the final regime. For the WKB treatment we have approximated the distribution function in the regions III and IV by the form (3.14) for the first order approximation. Looking at the shapes of the distribution function in fig. 2, we see that this approximation is not so satisfactory because there appears a "hill" around r0 (= 1 + 20 + ("(02)) and a step at ~ due to the approximation. The time (t = 4.0-6.0), around which the step is prominent, corresponds to the time at which the system begins to come

I ( ( ~ I ) ~)

a

I .o ~ - - -o !

0 5 ~ ~ ~ 05

[ ..... , ~ • _ . - - - ~ J 0 2 Z, 6 8

t

Fig . 3. T i m e e v o l u t i o n s o f t h e f i r s t a n d s e c o n d c u m u l a n t s o f r 2 f o r t h e s a m e p a r a m e t e r s a s in fig. 2 . I = 1121 a n d ( ( A / ) 2) = 1,4) (i(2))2.

WKB TREATMENT OF LASER RADIATION 87

into the final regime of the scaling theoryg). On the other hand there appears a "shoulder" around t =0.5-2.5 in the variance in fig. 3. This defect may come from the approximation of the summation of the eigenfunctions in (1.19) by the summation of the approximate eigenfunctions in (3.2). The terms which have negative contribution to the summation in (1.19) have been neglected in the approximate summation of (3.2). Then we may have overestimated the contribution to the moments from the regions I and II. This is the reason why we have the "shoulder" in the second cumulant (compare with fig. 2 of ref. 9).

By the comparison of the WKB treatment with the scaling treatmentg), we have obtained deeper understanding about the scaling theory. Furthermore, although there are some defects as are quoted above, we have got some informations about the extension of the scaling theory to the field theory in terms of the WKB approximation3). It is still an open problem.

Appendix A

Matching procedure

To relate the coefficients, Ku and Kh, of the wave function in the region (II), (2.6), with the coefficient, KI, of the wave function in the region (I), (2.3), we assume that in the matching region between (I) and (II) the potential V(r) in (2.6) can be approximated by V~(r), defined by (2.1). The wave function ~u(r) obtained by this approximation should be matched with the asymptotic, r ~o~, expression of (2.3). Using the asymptotic form of the Laguerre func- tion, L~)(z), for Izl large,

1 v)(_z)V{1 + 6(z_,)} L~)(z) r(1 +

+ ~ 1 ~ eZz-V-~{1 + ~(z-l)}, (A.I) I t - v )

the wave function Or(r) is evaluated asymptotically as

Ki~F~ e_~rz/4o~w~--:~[ 1 [rZV1 - - - I//I(r)~ [ - - F ( - - v ) \ ~ 80) ~ i

COS v'tr f r Z V 1 - 8 0 ) v } ,

e (r2120)~/"(~

(A.2)

for r ~ . Now we evaluate ~u(r) by approximating V(r) by Vl(r). The integral in the exponents of (2.6) becomes

88 T. ARIMITSU

r r

f _ _ f dovv,(x)- o x / V ( x ) - )~ = A~ oi c~

r

1 t 'dx , = ~-~ J --~--N/(1 - 80)x ' - 80~v/1 - 80(v + ½)x: - 02

a

' In[ r 2X/ i -80 ]~+~ V'I - 80 r2_ (v + ~ ) -

40 [ 0 V'-~v +- 1-~-+

7 / " + l g ( v ) - ~ + 0(0), (A.3)

where the first equality is the approximation of V(r) by Vl(r), (2.1), and ,~ by 2,v, (2.5), the third equality is obtained by neglecting the terms of the order of O, g(v) is defined by (2.9) and the relation V l (a )= 2~v has been used. In the matching region between (I) and (II), where V(r) >> ;~ (Vffr) >> ;~,), we have

0 0 20 + 0(02). (A.4) V" Vl(r) - )t,, X/-V~(r) r

Substituting (A.3) and (A.4) into (2.6) we get the expression

tOn(r) V r ~-~,:/4o~ ~-8o[./-4,.. [ e V l - 80 ]-~ ~ jg~_. /8/r ' \ ~ e i r 2 / 2 0 ) ~ ~ , V ~ / ~ ] I . . . . . . e - - tV(2~, + 1):+~] I,o,)

--- ,r eX/1-80 1 ~+~ (o) ~} + V 2 K i,[x/(~v + - ~ + ]1 e -jg'~'+~/8 . (1.5)

Matching the corresponding parts of (1.2) and (1.5), we obtain the relations (2.7) and (2.8).

As for the matching of qJjffr), (2.6), and Ore(r), (2.10), and that of +re(r) and +iv(r), (2.16), approximating the potential V(r) by Vm(r), (2.2), in qJH(r) and qJw(r) and using the asymptotic expansions of the Weber function, we obtain the relations between K, , Kh and Kin, Kh~ as (2.12) and (2.13), and those between KHb g~ll and Kiv as (2.17) and (2.18). We omit the detailed dis- cussions about them because they are given in appendix A of ref. 1).

Appendix B

Evaluat ion of m Ou.~(r)

The integral of the exponents in (2.20b), in which r is in the region (II), can be evaluated as follows. From (1.9) and (1.11), the potential V ( r ) = Vn=0(r) is

WKB TREATMENT OF LASER RADIATION 89

written in terms of U(r) as

V(r)= l { [u ' ( r ) ]~ -20 [ 1 U'(r )+ U"(r ) ] - -~} , (B.1)

where U ' ( r ) = dU(r ) /dr and U"(r)= d2U(r)/dr 2. Let ~/ be the point that for r <-0 the potential V(r) can be well approximated by V~(r), (2.1), and for r > "O in the region (II) where the relation U'(r) ,> U"(r) holds. Then for r > rl in the region (II)

1 0 d , X/V(r ) - )t ~ - ~ U'(r) + ~ ~rr[ln r + In U (r)] + ~ + ~(02), (B.2)

where we have used the fact that U' ( r )<O in this region, and we have neglected the terms of the order of 02 . The integral of the exponents can be evaluated by

- xp = - T V V ~ ( x ) - xp + V V ( x ) - ~ , (B.3) a a */

where the first term of the right-hand side is obtained by (A.3), putting r = in (A.3), and the integrand of the second term is approximated by (B.2). Finally we obtain

ex;/ = , , , ct

xexp(~0[U(r ) - U ( , ) I - ~ X / ] - Z - ~ ) [ r2eV~-- --80 ]P+½ OV(2p + 1) 2+I-I " (B.4)

Because V(r)~> •p in the region (II) we approximate in the normalization factor

1 X/V(r ) - Ap = - ~ U'(r) + t~(0). (B.5)

Substituting (B.4) and (B.5) into (2.20b), we obtain

(1) ~ 2 O i , . p ( r ) = e x p ( l [ u ( r ) - U (r/)] - ~ - - ~ / ] - - ~

1 - ,2)3,2v x ( ~ l ~ ( 1 - r2) 2 p ! [20(1 - r2) ] ' (B.6)

where we have neglected the term of the order of exp(-S,~(~p)).

90 T. ARIM1TSU

Acknowledgements

T h e a u t h o r w o u l d l ike to t h a n k P r o f e s s o r M. S u z u k i f o r u s e f u l a d v i c e and

c o m m e n t . M a n y s t i m u l a t i n g c o n v e r s a t i o n s w i t h c o l l e a g u e s o f t he K u b o -

W a d a - S u z u k i r e s e a r c h g r o u p at t he U n i v e r s i t y o f T o k y o a re a c k n o w l e d g e d .

References

1) B. Caroli, C. Caroli and B. Roulet, J. Stat. Phys. 21 (1979) 415. 2) B. Caroli, C. Caroli and B. Roulet, preprint. 3) M. Suzuki, Progr. Theor. Phys. 57 (1977) 477. 4) M. Suzuki, J. Stat. Phys. 16 (1977) 477. 5) M. Suzuki, Progr. Theor. Phys. 56 (1976) 77. 6) M. Suzuki, J. Stat. Phys. 16 (1977) 11. 7) H.J. de Vega, J.L. Gervais and B. Sakita, Nucl. Phys. B139 (1978) 20. 8) H. Risken, in: Progress in Optics, vol. VIII, E. Wolf, ed. (North-Holland, Amsterdam, 1970)

p. 239, and references therein. 9) T. Arimitsu and M. Suzuki, Physica 93A (1978) 574.

10) T. Arimitsu, Ph.D. Thesis, University of Tokyo (1980). 11) F. Haake, in: Springer Tracts in Modern Physics, vol. 66, G. H6hler, ed. (Springer,

Berlin-Heidelberg-New York, 1973) p. 98. 12) T. Arecchi et al., Phys. Rev. Lett. 16 (1966) 32. 13) T. Arecchi et al., Phys. Lett. 25A (1967) 59. 14) H. Risken and H. Vollmer, Z. Physik 201 (1967) 323. 15) H. Risken and H. Vollmer, Z. Physik 204 (1967) 240.